The Englishman is reportedly looking for a January move from the Emirates in order to gain more playing time.
Palace were reported as an interested suitor, but Hodgson has denied all knowledge of a potential move.
"Has he? Our scouting department look at everything, that's their job," Hodgson said when asked about the Welbeck rumours.
"Like all clubs in the Premier League, we are all constantly assessing and looking at players, so that when the day comes, if there is a vacancy and if there is money to spend, you know where to go.
"We aren't specifically targeting anything at this moment in time."
He continued: "But I must be perfectly honest with you and say that I haven't had a serious meeting with the scouting department - they are getting on with their work and we have several meetings scheduled later in November and December.
"Maybe then they will tell me what they are thinking in various positions, but if you ask me now: 'Have I spoken about Danny Welbeck or has anyone spoken to me about Danny Welbeck, the answer is no. So I have no idea where this has come from, you would have to ask the person who wrote it I guess."